$$$\ln^{5}\left(x\right)$$$ 的积分

求$$$\int \ln^{5}\left(x\right)\, dx$$$。

对于积分$$$\int{\ln{\left(x \right)}^{5} d x}$$$，使用分部积分法$$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

设 $$$\operatorname{u}=\ln{\left(x \right)}^{5}$$$ 和 $$$\operatorname{dv}=dx$$$。

则 $$$\operatorname{du}=\left(\ln{\left(x \right)}^{5}\right)^{\prime }dx=\frac{5 \ln{\left(x \right)}^{4}}{x} dx$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{1 d x}=x$$$ (步骤见 »)。

积分变为

$${\color{red}{\int{\ln{\left(x \right)}^{5} d x}}}={\color{red}{\left(\ln{\left(x \right)}^{5} \cdot x-\int{x \cdot \frac{5 \ln{\left(x \right)}^{4}}{x} d x}\right)}}={\color{red}{\left(x \ln{\left(x \right)}^{5} - \int{5 \ln{\left(x \right)}^{4} d x}\right)}}$$

对 $$$c=5$$$ 和 $$$f{\left(x \right)} = \ln{\left(x \right)}^{4}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$x \ln{\left(x \right)}^{5} - {\color{red}{\int{5 \ln{\left(x \right)}^{4} d x}}} = x \ln{\left(x \right)}^{5} - {\color{red}{\left(5 \int{\ln{\left(x \right)}^{4} d x}\right)}}$$

对于积分$$$\int{\ln{\left(x \right)}^{4} d x}$$$，使用分部积分法$$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

设 $$$\operatorname{u}=\ln{\left(x \right)}^{4}$$$ 和 $$$\operatorname{dv}=dx$$$。

则 $$$\operatorname{du}=\left(\ln{\left(x \right)}^{4}\right)^{\prime }dx=\frac{4 \ln{\left(x \right)}^{3}}{x} dx$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{1 d x}=x$$$ (步骤见 »)。

该积分可以改写为

$$x \ln{\left(x \right)}^{5} - 5 {\color{red}{\int{\ln{\left(x \right)}^{4} d x}}}=x \ln{\left(x \right)}^{5} - 5 {\color{red}{\left(\ln{\left(x \right)}^{4} \cdot x-\int{x \cdot \frac{4 \ln{\left(x \right)}^{3}}{x} d x}\right)}}=x \ln{\left(x \right)}^{5} - 5 {\color{red}{\left(x \ln{\left(x \right)}^{4} - \int{4 \ln{\left(x \right)}^{3} d x}\right)}}$$

对 $$$c=4$$$ 和 $$$f{\left(x \right)} = \ln{\left(x \right)}^{3}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 5 {\color{red}{\int{4 \ln{\left(x \right)}^{3} d x}}} = x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 5 {\color{red}{\left(4 \int{\ln{\left(x \right)}^{3} d x}\right)}}$$

对于积分$$$\int{\ln{\left(x \right)}^{3} d x}$$$，使用分部积分法$$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

设 $$$\operatorname{u}=\ln{\left(x \right)}^{3}$$$ 和 $$$\operatorname{dv}=dx$$$。

则 $$$\operatorname{du}=\left(\ln{\left(x \right)}^{3}\right)^{\prime }dx=\frac{3 \ln{\left(x \right)}^{2}}{x} dx$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{1 d x}=x$$$ (步骤见 »)。

该积分可以改写为

$$x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 {\color{red}{\int{\ln{\left(x \right)}^{3} d x}}}=x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 {\color{red}{\left(\ln{\left(x \right)}^{3} \cdot x-\int{x \cdot \frac{3 \ln{\left(x \right)}^{2}}{x} d x}\right)}}=x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 {\color{red}{\left(x \ln{\left(x \right)}^{3} - \int{3 \ln{\left(x \right)}^{2} d x}\right)}}$$

对 $$$c=3$$$ 和 $$$f{\left(x \right)} = \ln{\left(x \right)}^{2}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 x \ln{\left(x \right)}^{3} - 20 {\color{red}{\int{3 \ln{\left(x \right)}^{2} d x}}} = x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 x \ln{\left(x \right)}^{3} - 20 {\color{red}{\left(3 \int{\ln{\left(x \right)}^{2} d x}\right)}}$$

对于积分$$$\int{\ln{\left(x \right)}^{2} d x}$$$，使用分部积分法$$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

设 $$$\operatorname{u}=\ln{\left(x \right)}^{2}$$$ 和 $$$\operatorname{dv}=dx$$$。

则 $$$\operatorname{du}=\left(\ln{\left(x \right)}^{2}\right)^{\prime }dx=\frac{2 \ln{\left(x \right)}}{x} dx$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{1 d x}=x$$$ (步骤见 »)。

所以，

$$x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 x \ln{\left(x \right)}^{3} - 60 {\color{red}{\int{\ln{\left(x \right)}^{2} d x}}}=x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 x \ln{\left(x \right)}^{3} - 60 {\color{red}{\left(\ln{\left(x \right)}^{2} \cdot x-\int{x \cdot \frac{2 \ln{\left(x \right)}}{x} d x}\right)}}=x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 x \ln{\left(x \right)}^{3} - 60 {\color{red}{\left(x \ln{\left(x \right)}^{2} - \int{2 \ln{\left(x \right)} d x}\right)}}$$

对 $$$c=2$$$ 和 $$$f{\left(x \right)} = \ln{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 x \ln{\left(x \right)}^{3} - 60 x \ln{\left(x \right)}^{2} + 60 {\color{red}{\int{2 \ln{\left(x \right)} d x}}} = x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 x \ln{\left(x \right)}^{3} - 60 x \ln{\left(x \right)}^{2} + 60 {\color{red}{\left(2 \int{\ln{\left(x \right)} d x}\right)}}$$

对于积分$$$\int{\ln{\left(x \right)} d x}$$$，使用分部积分法$$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

设 $$$\operatorname{u}=\ln{\left(x \right)}$$$ 和 $$$\operatorname{dv}=dx$$$。

则 $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{1 d x}=x$$$ (步骤见 »)。

积分变为

$$x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 x \ln{\left(x \right)}^{3} - 60 x \ln{\left(x \right)}^{2} + 120 {\color{red}{\int{\ln{\left(x \right)} d x}}}=x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 x \ln{\left(x \right)}^{3} - 60 x \ln{\left(x \right)}^{2} + 120 {\color{red}{\left(\ln{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{x} d x}\right)}}=x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 x \ln{\left(x \right)}^{3} - 60 x \ln{\left(x \right)}^{2} + 120 {\color{red}{\left(x \ln{\left(x \right)} - \int{1 d x}\right)}}$$

应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=1$$$：

$$x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 x \ln{\left(x \right)}^{3} - 60 x \ln{\left(x \right)}^{2} + 120 x \ln{\left(x \right)} - 120 {\color{red}{\int{1 d x}}} = x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 x \ln{\left(x \right)}^{3} - 60 x \ln{\left(x \right)}^{2} + 120 x \ln{\left(x \right)} - 120 {\color{red}{x}}$$

因此，

$$\int{\ln{\left(x \right)}^{5} d x} = x \ln{\left(x \right)}^{5} - 5 x \ln{\left(x \right)}^{4} + 20 x \ln{\left(x \right)}^{3} - 60 x \ln{\left(x \right)}^{2} + 120 x \ln{\left(x \right)} - 120 x$$

化简：

$$\int{\ln{\left(x \right)}^{5} d x} = x \left(\ln{\left(x \right)}^{5} - 5 \ln{\left(x \right)}^{4} + 20 \ln{\left(x \right)}^{3} - 60 \ln{\left(x \right)}^{2} + 120 \ln{\left(x \right)} - 120\right)$$

加上积分常数：

$$\int{\ln{\left(x \right)}^{5} d x} = x \left(\ln{\left(x \right)}^{5} - 5 \ln{\left(x \right)}^{4} + 20 \ln{\left(x \right)}^{3} - 60 \ln{\left(x \right)}^{2} + 120 \ln{\left(x \right)} - 120\right)+C$$

$$$\int \ln^{5}\left(x\right)\, dx = x \left(\ln^{5}\left(x\right) - 5 \ln^{4}\left(x\right) + 20 \ln^{3}\left(x\right) - 60 \ln^{2}\left(x\right) + 120 \ln\left(x\right) - 120\right) + C$$$A